To solve the problem, we need to find the probability of three different events related to the page numbers of the NCERT class IX Mathematics book, which contains a total of 323 pages.
### Step 1: Understanding Probability
The probability of an event is given by the formula:
\[
P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}}
\]
In this case, the total outcomes are the total number of pages, which is 323.
### Step 2: (i) Probability that the page number contains 9 at one's place
We need to find how many page numbers from 1 to 323 have 9 at the one's place.
- The page numbers that have 9 at the one's place are: 9, 19, 29, 39, 49, 59, 69, 79, 89, 99, 109, 119, 129, 139, 149, 159, 169, 179, 189, 199, 209, 219, 229, 239, 249, 259, 269, 279, 289, 299, 309, 319.
- Counting these, we find there are 32 favorable outcomes.
Now we can calculate the probability:
\[
P(\text{9 at one's place}) = \frac{32}{323}
\]
### Step 3: (ii) Probability that the page number is a multiple of 4
Next, we need to find how many page numbers from 1 to 323 are multiples of 4.
- The multiples of 4 can be found by dividing the total pages by 4. The largest multiple of 4 less than or equal to 323 is 320.
- The multiples of 4 from 1 to 320 are: 4, 8, 12, ..., 320.
- This forms an arithmetic sequence where the first term \(a = 4\) and the common difference \(d = 4\).
- The number of terms \(n\) can be calculated using the formula for the nth term of an arithmetic sequence:
\[
n = \frac{\text{last term} - \text{first term}}{d} + 1 = \frac{320 - 4}{4} + 1 = 80
\]
Now we can calculate the probability:
\[
P(\text{multiple of 4}) = \frac{80}{323}
\]
### Step 4: (iii) Probability that the page number is a perfect square
Lastly, we need to find how many page numbers from 1 to 323 are perfect squares.
- The perfect squares less than or equal to 323 are: \(1^2, 2^2, 3^2, ..., 18^2\) (since \(18^2 = 324\) is greater than 323).
- The perfect squares are: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324 (but we exclude 324).
- Counting these, we find there are 17 favorable outcomes.
Now we can calculate the probability:
\[
P(\text{perfect square}) = \frac{17}{323}
\]
### Final Answers:
1. Probability that the page number contains 9 at one's place: \( \frac{32}{323} \)
2. Probability that the page number is a multiple of 4: \( \frac{80}{323} \)
3. Probability that the page number is a perfect square: \( \frac{17}{323} \)
